Nonuniform Sampling Theorem for Non-decaying Signals in Mixed-Norm Spaces \(L_{\vec{p},\frac{1}{\omega }}(\mathbb{R}^{d})\)

Abstract

In this paper, combining the non-decaying properties with the mixed-norm properties, the revelent sampling problems are studied under the target space of \(L_{\vec{p},\frac{1}{\omega }}(\mathbb{R}^{d})\). Firstly, we will give a stability theorem for the shift-invariant subspace \(V_{\vec{p},\frac{1}{\omega }}(\varphi )\). Secondly, an ideal sampling in \(W_{\vec{p},\frac{1}{\omega }}^{s}(\mathbb{R}^{d})\) is proved, and thirdly, a convergence theorem (or algorithm) is shown for \(V_{\vec{p},\frac{1}{\omega }}(\varphi )\). It should be pointed out that the auxiliary function \(\varphi \) enjoys the membership in a Wiener amalgam space.